\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
\( \frac{w_{mn}}{w_{nm}} = \frac{p^\star_m}{p^\star_n} = \exp\left(-\frac{\epsilon_m-\epsilon_n}{T} \right)\)
\( \dot{S} \geq \frac{\dot{Q}}{T} \)
\( S(P) = f\left(\sum_m g(p_m) \right) \)
Maximize S(p) subject to constraint that p is normalized and expected energy has a given value
Solution: MaxEnt distribution: \( p^\star_m = (g')^{-1} \left(\frac{\alpha+\beta \epsilon_m}{C_f} \right) \), \( C_f = f'(\sum_m g(p_m)) \)
\( \)
\( U = \sum_m p_m \epsilon_m\)
\( S = f\left(\sum_m g(p_m) \right) \)
\( S = -\sum_m p_m \log p_m \)
\( \dot{p}_m = \sum_n \left[J(w_{mn},p_n)-J(w_{nm},p_m) \right] \)
\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
\( \sum_m \dot{p}_m = 0\)
\( w_{mn}\)
\( \left[J(w_{mn},p_n)-J(w_{nm},p_m) \right]\)
\(p^\star_m = (g')^{-1} \left( \frac{\alpha+\beta \epsilon_m}{C_f} \right) \)
\(p^\star_m = \exp(-\alpha-\beta \epsilon_m) \)
\( J(w_{mn},p^\star_n)=J(w_{nm},p^\star_m) \)
\(w_{mn} p^\star_n = w_{nm} p^\star_m\)
\(\frac{\mathrm{d} S}{\mathrm{d} t} = \dot{S}_i + \dot{S}_e\)
\( \dot{S}_i \geq 0\) and \(\dot{S}_i = 0 \Leftrightarrow J(w_{mn},p_n) = J(w_{nm},p_m) \ \forall \ m,n\)
\( \dot{S}_e = \frac{1}{T} \sum_m \dot{p}_m \epsilon_m = \frac{\dot{Q}}{T} \)
\(J(w_{mn},p_n) = \psi( j(w_{mn}) - g'(p_n)) \)
\(j\) - arbitrary function
\(\psi\) - increasing function
\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
\(J_{mn} = w_{mn} p_n = \exp(\log w_{mn} + \log p_n)\)
\(\Rightarrow g'(p_n) = - \log(p_n) \)
\( \Rightarrow S = - \sum_n p_n \log p_n\)
\(H = H_{system} +H_{bath} \)
\( \lambda \, H_{bath}(x_1,\dots,x_n) = H_{bath}( \lambda^{1/a_1} x_1,\dots, \lambda^{1/a_n} x_n) \)
\( p(E) \propto \int \delta(E - H_{bath}) \, \mathrm{d} x_1 \dots \mathrm{d} x_n \)
\( p(E) \propto (1-(q-1) \beta E)^{1/(q-1)} \)
\( S = \frac{1}{1-q} (\sum_m p_m^q-p_m)\)
\( \Rightarrow g'(p_m) = \frac{q p_m^{q-1}-1}{1-q}\)
\( J_{mn} = \psi(j(w_{mn}) + \frac{q p_m^{q-1}-1}{q-1} )\)
\( \dot{S} = - \sum_m \dot{p}_m \log p_m \)
\(= - \frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{p_m}{p_n}\)
\(= \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} p_n}{w_{nm} p_m}}_{\dot{S}_i}\)
\(+ \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} }{w_{nm} }}_{\dot{S}_e}\)
\(= \dot{S}_i + \dot{S}_e \geq \frac{\dot{Q}}{T}\)
\( \dot{S} = C_f \sum_m \dot{p}_m g'(p_m) \)
\(= \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m) - g'(p_n)) \)
\( = \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (\Phi_{mn} - \Phi_{nm}) \)
\(+ \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m)+\Phi_{nm} - g'(p_n) - \Phi_{mn}) \)
\( = \underbrace{\frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (\phi(J_{mn}) - \phi(J_{nm}) )}_{\dot{S}_i}\)
\(+ \underbrace{\frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m)+\phi(J_{nm}) - g'(p_n) - \phi(J_{mn}) )}_{\dot{S}_e} \)
\(\dot{S}_i \Rightarrow \phi - \ increasing\)
\(\dot{S}_e \Rightarrow C_f[g'(p_m) + \phi(J_{nm}) - g'(p_n) - \phi(J_{mn})] = \frac{\epsilon_n - \epsilon_m}{T} \)
\(\Rightarrow \phi(J_{mn}) = j(w_{mn}) - g'(p_n) \)
\(\Rightarrow J_{mn} = \psi(j(w_{mn}) - g'(p_n))\), \( \psi = \phi^{-1} \) - increasing \( \square . \)
\( j(w_{mn}) - j(w_{nm}) = \frac{\epsilon_n - \epsilon_m}{C_f T} \)
\( \beta = \frac{1}{T} \)